I thought about entering this in the napkin sketch contest, but I don't have any need for a new case and it seemed a bit more serious than the usual fare there. Not serious enough that I'll likely bother building it, but serious enough that it deserves its own thread, I think.
The basic idea is to get a floor-standing tower, maybe 7.5" ID and 4+ feet tall, leave the top and bottom open to airflow, and build a computer in the bottom of it. The more heat the components produce, the more air passively flows up the tower. In a simple form, it should be able to replace case fans pretty well. In a more advanced design, it could be ducted so that all the air has to flow through the CPU/GPU heatsinks, potentially resulting in a mostly passive yet very powerful system.
When thinking of more specific layouts, I can't quite seem to hit all the design points I want. They are (in no particular order): CPU and GPU as low as possible for maximum space efficiency, PSU and SSDs below CPU and GPU for better reliability, filtered air intake, ability to fit full-length graphics cards, easy access to ports, compact but stable (probably round) footprint, and support for ducted cooling of individual components. I've got designs that can hit just about any five of those seven points at once. My favorite so far hits everything except the easy port access:
That's with a PCI-e extender to wrap the graphics card around behind the mobo, like in the A4-SFX. I think that's about as good as cooling is going to get in that form factor, especially since the graphics card can be isolated from everything else. The trouble is that all the ports are upward-facing, so the chimney will have to be detachable somehow and have a hole in its side for cables. That hole will have to have some kind of grommet to block most airflow. It also has some requirements for the CPU and GPU coolers, though I don't think they're particularly onerous ones. (A blower GPU cooler would probably work better, but doesn't seem essential.) If it's all working properly and using conventional fan controls, it'll probably have normal amounts of fan noise when load first shows up, but when under a sustained load like gaming it should drop to very low noise as passive airflow through the heatsinks picks up (assuming fan controls are set up suitably). Quicker response should be achievable by narrowing the inside of the tower just above the cable routing area, and not expanding it to full width again until a ways up.